Reverse quantum annealing of the $p$-spin model with relaxation

In reverse quantum annealing, the initial state is an eigenstate of the final problem Hamiltonian and the transverse field is cycled rather than strictly decreased as in standard (forward) quantum annealing. We present a numerical study of the reverse quantum annealing protocol applied to the $p$-spin model ($p=3$), including pausing, in an open-system setting accounting for dephasing in the energy eigenbasis, which results in thermal relaxation. We consider both independent and collective dephasing and demonstrate that in both cases the open-system dynamics substantially enhances the performance of reverse annealing. Namely, including dephasing overcomes the failure of purely closed-system reverse annealing to converge to the ground state of the $p$-spin model. We demonstrate that pausing further improves the success probability. The collective dephasing model leads to somewhat better performance than independent dephasing. The protocol we consider corresponds closely to the one implemented in the current generation of commercial quantum annealers, and our results help to explain why recent experiments demonstrated enhanced success probabilities under reverse annealing and pausing.

Figure 1

(a) Annealing schedules (in units such that $\hslash =1$) as a function of the annealing fraction $s\left(t\right)$, chosen to be similar to the schedules of the D-Wave processors. (b) Annealing fraction $s\left(t\right)$. The blue solid curve represents standard forward quantum annealing of total annealing time $\tau =500$ ns. The red dashed curve represents reverse annealing of the same total duration with an inversion point $(t_{\text{inv}}=200\mathrm{ns},s_{\text{inv}}=0.6)$.

Figure 2

(a) Success probability in unitary reverse annealing as a function of the inversion point $s_{\text{inv}}$ for several values of the magnetization of the initial state. The dashed vertical line indicates $s_{\text{inv}}=s_{\mathrm{\Delta }}\approx 0.309$. The annealing time is $\tau =100$ ns. We sampled the interval $s_{\text{inv}}\in (0,1)$ using a step size of $\mathrm{\Delta }s=0.002$ and repeated the dynamics for each choice of $s_{\text{inv}}$. (b) Maximum success probability achievable with unitary reverse annealing as a function of the magnetization of the initial state for two annealing times $\tau =100$ ns and $\tau =1000$ ns. The inset zooms in on the region $m\in [-1.0,0.0]$. Note the logarithmic scale on the vertical axis.

Figure 3

Success probability in reverse annealing as a function of the inversion point $s_{\text{inv}}$ for several values of the magnetization of the initial state. The $p$-spin system is coupled to a collective dephasing bosonic environment as in Eq. (10), and the coupling strength is $\eta =1\times {10}^{-3}$. The dashed vertical line denotes the time $s_{\mathrm{\Delta }}$ of the avoided crossing between the ground state and the first excited state. In (a) the annealing time is $\tau =100$ ns, in (b) $\tau =500$ ns. We sampled the interval $s_{\text{inv}}\in (0,1)$ using a step $\mathrm{\Delta }s=0.02$. All other parameters are given in the main text.

Figure 4

Success probability in reverse annealing as a function of the inversion point $s_{\text{inv}}$ for $n\in \{3,\cdots ,8\}$. The initial state is the first excited state of the maximum spin subspace ($m_0=1-2/n$). The dashed vertical line denotes the time $s_{\mathrm{\Delta }}$ of the avoided crossing between the ground state and the first excited state. The annealing time is $\tau =100$ ns. We sampled the interval $s_{\text{inv}}\in (0,1)$ using a step size of $\mathrm{\Delta }s=0.005$.

Figure 5

Maximum success probability achievable with reverse annealing as a function of the number of qubits $n$ using the collective and the independent dephasing models of Eqs. (10) and (12), respectively. The annealing time is $\tau =100$ ns.

Figure 6

Success probability in paused reverse annealing for the collective dephasing model as a function of the inversion point $s_{\text{inv}}$ for several values of the magnetization of the initial state. The coupling strength is $\eta =1\times {10}^{-3}$. The annealing time is $\tau =100$ ns. In (a) a pause of duration $t_{\text{p}}=100$ ns is inserted at the inversion point, while in (b) $t_{\text{p}}=400$ ns. All other parameters are given in the main text. Dotted lines with empty symbols refer to open-system reverse annealing of time $\tau =100$ ns and no pauses. The dashed vertical line denotes $s_{\text{inv}}=s_{\mathrm{\Delta }}$. The interval $s_{\text{inv}}\in (0,1)$ is sampled using a step size $\mathrm{\Delta }s=0.02$.

Figure 7

Comparison of the success probability in reverse annealing for the collective and independent dephasing models as a function of the inversion point $s_{\text{inv}}$ for $n\in \{3,\cdots ,8\}$. The initial state is the first excited state of the maximum spin subspace ($m_0=1-2/n$). The dashed vertical line denotes the time $s_{\mathrm{\Delta }}$ of the avoided crossing between the ground state and the first excited state. The annealing time is $\tau =100$ ns, and a pause of duration $t_{\text{p}}=100$ ns is inserted at the inversion point.